Interstitial diffusion and the relationship between compartment modelling and multi-scale spatial-temporal modelling...

Tumour cell proliferation can be imaged via positron emission tomography of the radiotracer 3'-deoxy-3'-18F-fluorothymidine (18F-FLT). Conceptually, the number of proliferating cells might be expected to correlate more closely with the kinetics of 18F-FLT uptake than with uptake at a fixed time. Radiotracer uptake kinetics are standardly visualized using parametric maps of compartment model fits to time-activity-curves (TACs) of individual voxels. However the relationship between the underlying spatiotemporal accumulation of FLT and the kinetics described by compartment models has not yet been explored. In this work tumour tracer uptake is simulated using a mechanistic spatial-temporal model based on a convection-diffusion-reaction equation solved via the finite difference method. The model describes a chain of processes: the flow of FLT between the spatially heterogeneous tumour vasculature and interstitium; diffusion and convection of FLT within the interstitium; transport of FLT into cells; and intracellular phosphorylation. Using values of model parameters estimated from the biological literature, simulated FLT TACs are generated with shapes and magnitudes similar to those seen clinically. Results show that the kinetics of the spatial-temporal model can be recovered accurately by fitting a 3-tissue compartment model to FLT TACs simulated for those tumours or tumour sub-volumes that can be viewed as approximately closed, for which tracer diffusion throughout the interstitium makes only a small fractional change to the quantity of FLT they contain. For a single PET voxel of width 2.5-5 mm we show that this condition is roughly equivalent to requiring that the relative difference in tracer uptake between the voxel and its neighbours is much less than one.